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In mathematical writing, I've often seen people use the expression 'Then are equivalent' to introduce a list of conditions that are logically equivalent to each other (and I've used it myself a few times). E.g., here is an excerpt from the bottom of p. 6 (right column) in a 2015 open-access article from Nature Communications (I'm not sure about the nationality of the authors, though):

Proposition II. [...]. Then are equivalent

(1) For a large enough $A$, and corresponding $A'$, there exists a trace-preserving unital map [...] such that [...].

(2) For a large enough $B$, and large enough $B'$, there exists a trace-nonincreasing subunital map [...] such that [...].

And here is another example, from p. 253 in a paper of W.-D. Heinrichs (a German scholar) appeared in Publ. RIMS, Kyoto Univ., Vol. 33 (1997), 241-255:

Theorem 4.8. [...]. Then are equivalent:

  1. [...] satisfies the density condition (DC).

  2. [...] satisfies the strong dual density condition by operator (SDDCO).

  3. [...] is a bornological (DFO)-space.

But I'm a non-native speaker, and a colleague from the US has recently questioned the correctness of the expression, arguing that he finds it "strange, and possibly ungrammatical."

I'd say that the expression is not so uncommon in (mathematical) papers authored or co-authored by native speakers, but I don't have sufficiently robust statistics to advocate for it. So I'd appreciate to hear from the forum on this issue.

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    adv-V-C-S is a very rare sentence structure. I believe it's grammatical (usage determines this in the final analysis) in mathspeak, but I'd avoid it like the plague elsewhere. I'd suggest you ask this on Mathematics.SE, as mathspeak and standard English don't overlap fully. – Edwin Ashworth Aug 4 at 13:43
  • @EdwinAshworth I'll wait a little bit and then ask on Mathematics.SE as per your suggestion (unless I get one or more answers here). – Salvo Tringali Aug 4 at 14:29
  • I'll add that the construction is probably not ungrammatical. But using something like 'Then are resplendent the osprey and the peregrine' belongs in poetry if anywhere. In everyday speech or writing, they're unacceptable on grounds of unidiomaticity. – Edwin Ashworth Aug 4 at 14:40
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    I don't think I've ever seen this expression before. Without context, it looks nonsensical at best to me. I can't say if it's actually ungrammatical, as it's being used absent a complete sentence or context. Can you provide at least one example of its use? – Jason Bassford Aug 4 at 16:27
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    Going by the names alone, I would guess two of the authors of the first article are German and the other two French. This word order would be formal but perfectly grammatical and not uncommon in German, and though definitely uncommon in French, I do think it would be grammatical in formal writing. A working hypothesis would be that this is Denglish, German features spilling affecting how Germans speak and write English. – Janus Bahs Jacquet Aug 4 at 23:03
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As a mathematician, I find your expression looks odd, as if translated literally from some other language. The word 'Then' is normally used to introduce some statement that follows from preceding reasoning, as in 'if 1 and 1 make 2, then...'. A sentence beginning with 'Then ' is weird.

If you mean 'from the preceding material we conclude that the following statements are equivalent' then say so. If, on the other hand yo are stating a theorem, you don't need 'Then'; just say 'The following are equivalent:'

  • What you say is true, but note that it’s Proposition II and Theorem 4.8. The word then is clearly being used to express dependency on a previous statement. It’s pretty terse, though. – Global Charm Aug 5 at 15:31
  • The use of 'Then the following hold:' and the like to introduce a list of conclusions drawn from some premises in the statement of lemmas, propositions, theorems, etc. is standard in mathematical writing. But I'm finally convinced that the expression "Then are equivalent:" is probably a literal translation of one commonly used in other languages (e.g., in German), as also suggested by @JanusBahsJacquet in a comment to the OP. – Salvo Tringali Aug 8 at 21:34

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